Two-species system with nonlocal interactions driven by Riesz potentials

TL;DR

This paper studies a two-species nonlocal interaction model using Riesz potentials, establishing weak solutions through optimal transport and gradient flow techniques, extending prior results to more singular and coupled interactions.

Contribution

It introduces a novel analysis for two-species systems with possibly singular, fractional Riesz interactions, broadening the scope of existing models.

Findings

Proved existence of weak solutions under singularity conditions.

Extended analysis to systems with different fractional orders in interactions.

Handled non-symmetric cross-interaction kernels.

Abstract

This paper investigates a system of nonlocal continuity equations modelling the interaction of two species coupled through Riesz-type potentials. The model incorporates self- and cross-interaction kernels of possibly different fractional orders. By exploiting optimal transportation theory and the theory of gradient flows in Wasserstein spaces, we establish the existence of weak solutions under singularity assumptions on all interaction potentials, provided the cross-interaction ones satisfy a symmetry condition. Our analysis extends previous results available for either single-species equations or multi-species systems with smoother cross-interaction kernels.